12881289

Remembering W這dek Ko這s
by C.C.J. Roothaan
HewlettPackard Laboratories, Palo Alto, California 94304, USA Professor Emeritus of Physics and Chemistry, University of Chicago, Chicago, Illinois 60637, USA
(Received August 18th, 1997; accepted August 19th, 1997)

12901292

W這dzimierz Ko這s at the Laboratory of Molecular Structure and Spectra, 19571968: A Brief Memoir
by B.J. Ransil
Departments of Medicine, Beth Israel Hospital and Harvard Medical School, (Ret.), Department of Neurology, Beth Israel Deaconess Medical Center, Boston, Massachusetts, U.S.A
(Received April 17th, 1997; revised May 30th; 1997, accepted July 15th, 1997)

12931303

Understanding the Adiabatic Approximation; the Accurate Data of H_{2} Transferred to H_{3}^{+}
by J. Hinze, A. Alijah and L. Wolniewicz
Fakultaet fuer Chemie, Universitaet Bielefeld, 33615 Bielefeld, Germany Institute of Physics, Nicholas Copernicus University, Torun, Poland
(Received October 6th, 1997; revised October 8th, 1997; accepted October 11th, 1997)
Different ab initio calculations of the rotationvibrational states of H_{3}^{+} and its isotopomers are reviewed. In the critical discussion of the results obtained by various authors, we have attempted to analyse the origin of the slight differences remaining (i) due to the inadequacies in the potential surface or (ii) due to the adiabatic approximation used

13041312

Total Energies and Equilibrium Bond Lengths of Molecules: Myths or Realities, the Example of the Hydrogen Molecule
by J.M. Andre
Laboratoire de Chimie Theorique Appliquee, Facultes Universitaires NotreDame de la Paix Rue de Bruxelles, 61 B5000NAMUR, Belgium
(Received March 27th, 1997; revised December 19th, 1997; accepted December 20th, 1997)
In this paper, simple considerations of use in everyday quantum chemistry are reviewed. They stress the fact that the "classical" concepts of total energy and of bond lengths in molecules are based on severe approximations and are not as such accessible to direct experimentation, due to Heisenberg's uncertainty principle and to its main consequence in molecules: the existence of a residual zero point vibration. In this paper, we address successively (a) the determination of total energies, and (b) the meaning of an "equilibrium" bond length.

13131322

The SchroedingerRiccati Equation. An Application to the Hydrogen and Helium Atoms
by S. Fraga and E.S. Fraga
Department of Chemistry, University of Alberta, Edmonton, Alberta T6G 2G2, Canada Department of Chemical and Biochemical Engineering, University College London, Torrington Place, London WC1E 7JE, U.K.
(Received March 25th, 1997; revised May 25th, accepted June 1st, 1997)
Numerical results for the ground states of the hydrogen and helium atoms confirm the validity of the algebraic SchroedingerRiccati equation. A quartic equation may be appropriate for points in the electron configuration space corresponding to electron positions close to the nucleus but a higher degree will be needed for the outer regions. The SchroedingerRiccati equation is appropriate for implementation in parallelcomputing facilities and its practical applicability will be enhanced by the availability of highspeed computers.

13231333

NonAdiabatic Couplings between the Final States of Tritium Beta Decay
by S. Jonsell^{1}, A. Saenz^{1,2} and P. Froelich^{1}
^{1}Department of Quantum Chemistry, Uppsala University, Box 518, S75120 Uppsala, Sweden
^{2}Fakultaet fuer Chemie, Universitaet Konstanz, Postfach 5560 M722, D78434 Konstanz, Germany
(Received April 3rd, 1997; revised June 26th, 1997; accepted June 30th, 1997)
The nonadiabatic couplings between the ground state and the first 5 excited states in ^{3}HeT^{+} have been calculated. Nonadiabatic corrections to the ßdecay spectrum of T_{2} are derived in firstorder perturbation theory. These corrections are estimated to be less than 0.01%.

13341342

General Formula for ShapeResonance Life Times: The HeH^{+} Example
by J.M. Peek
45 Puye, Los Alamos, New Mexico 87544, U.S.A.
(Received April 8th, 1997; revised June 4th, 1997; accepted June 15th, 1997)
Translational spectroscopy stimulated the investigation of shape resonances in the electronic ground state of HeH^{+} a number of years ago. Subsequent spectroscopic studies have increased the accuracy of these experimental data. Theoretical investigations require an extremely accurate potential energy curve over an extended range of internuclear separation. The calculation by Ko這s proved invaluable in identifying these states and continues to be the standard to which improvements are added. The accurate characterization of many isolated shape resonances may be necessary so convenient techniques are required. Test functions are often used to create a boundary condition that converts this problem in the continuum to a discrete state calculation. The resulting eigenvalue search is efficient and convenient to automate. The associated widths are usually found by other techniques. It is shown here that the eigenvalue calculation implies a width if one convenient and easily satisfied requirement is met. Two versions of this result are applied to selected experimental data for HeH^{+}.

13431352

Quadrupole Splitting of the Weakly Bound System [(dtµ)
_{11}dee]
by D. Bakalov^{1}, K. Bakalova^{2}, V. Korobov^{3} and H.J. Monkhorst^{4}
^{1}Institute for Nuclear Research and Nuclear Energy, Sofia, Bulgaria
^{2}SolarTerrestrial Influences Laboratory, Sofia, Bulgaria
^{3}JINR, Dubna, Russia
^{4}QTP, University of Florida, Gainesville, FL32611
(Received April 30th, 1997; revised May 19th, 1997; accepted May 30th, 1997)
The energy levels of the molecular complex [(dtµ)_{11}dee] are close to the sum of the energies of the (dtµ)_{11} and D_{2} molecules; the small difference of a few meV, referred to as "finite size correction", is an effect of the Coulomb interaction with the spatially distributed electrical charge in the compound nucleus of the complex. The quadrupole terms in the multipole expansion of the latter are shown to interfere with the spin interactions and to modify the hyperfine structure of the levels of the complex in a way that may affect the resonant formation rate of (dtµ)_{11}.

13531360

The Ground State of (HeHHe)^{+} from Correlated Ab Initio Calculations
by J. Komasa^{1} and J. Rychlewski^{2}
^{1}Department of Chemistry, A. Mickiewicz University, Grunwaldzka 6, 60780 Poznan, Poland ^{2}Department of Chemistry, A. Mickiewicz University, Grunwaldzka 6, 60780 Poznan, Poland and Poznan Supercomputing and Networking Center, Wieniawskiego 17/19, 61713 Poznan, Poland
(Received April 27th, 1997; accepted June 30th, 1997)
He_{2}H^{+} ion in its ground state is studied by means of ab initio methods taking into account the electron correlation. Geometry optimization at the CCSD(T)/ccpV5Z level of theory has been performed and the potential energy hypersurface scan is presented. The optimum conformation is linear and symmetric with the proton at the HeHe midpoint and the HeH distance equal to 1.75 bohr. Stabilization energy with respect to the He_{2}H^{+} HeH^{+} + He dissociation channel has been computed. Additionally, a single point variational calculations with the use of the Exponentially Correlated Gaussian wave functions have been performed. They supply an upper bound to both the total electronic energy (5.903505 hartree) and the stabilization energy (13.224 kcal/mol). A comparison of the results from both the perturbational coupled cluster and variational methods is presented.

13611375

A Boundary Condition Determined Wave Function for the H_{2} (X^{1}_{g}) Molecule
by U. Kleinekathoefer^{1,2}, S.H. Patil^{1,3}, K.T. Tang^{1,4} and J.P. Toennies^{1}
^{1}MaxPlanckInstitut fuer Stroemungsforschung, Bunsenstr.10, D37073 Goettingen, Germany
^{2}Department of Chemical Physics, Weizmann Institute of Science, Rehovot 76100, Israel
^{3}Department of Physics, Indian Institute of Technology, Bombay 400076, India
^{4}Institute of Atomic and Molecular Sciences, Academica Sinica, P.O.Box 23166, Taipei, Taiwan 106, Republic of China
(Received April 9th, 1997; revised May 24th, 1997; accepted June 3rd, 1997)
Two relatively simple nonvariational wave functions for two electron diatomic molecules are proposed. The electronelectron cusp condition is satisfied rigorously by a correlation function which has the correct behavior for r_{12} 0 and r_{12} . The electronnucleus cusp conditions are also rigorously satisfied by the proposed functional forms of the wave functions. The parameters are chosen to match the asymptotic conditions in two different approximations. Both wave functions yield very good energies for the chemical bond. This demonstrates that these local conditions have a big effect on the wave function.

13761388

Polarization Propagator Calculations of Molecular Generalized Oscillator Strengths
by J.R. Sabin and J. Oddershede
Kemisk Institut, Odense Universitet, 5230 Odense M, Denmark Quantum Theory Project, Department of Physics, University of Florida, Gainesville, FL 32611 U.S.A.
(Received May 21th, 1997; revised July 5th, 1997; accepted July 15th, 1997)
The linear energy deposition, or stopping power, of materials for swift ions is determined, at the level of the first Born approximation, by the generalized oscillator strength distribution of the target. Here we consider the calculation of generalized oscillator strengths for several low lying transitions in small molecules using the polarization propagator approach, with emphasis on their directional characteristics.

13891397

Adiabatic Separations of the Vibrational Motions in HeH_{2}^{+} and the Calculation of Metastable States
by V. Spirko^{1}, M. Jurek^{1} and W.P. Kraemer^{2}
^{1}J. Heyrovsky Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, Dolejskova 3, CZ18223 Praha 8, Czech Republic
^{2}MaxPlanckInstitute of Astrophysics, Postfach 1523, D85740 Garching, Germany
(Received June 8th, 1997; accepted July 5th, 1997)
Recent ab initio calculated potential energy data for the ground electronic state of HeH_{2}^{+} are used to construct a new potential function, which is suitable to describe the interaction between He and H_{2}^{+} at low collision energies. All bound and the lowest quasibound rotationvibrational levels of HeH_{2}^{+} are evaluated within the framework of the SutcliffeTennyson Hamiltonian for triatomic molecules neglecting Coriolis interactions between the states. Adiabatic separations of the vibrational motions are used to simplify the calculation of the bound and quasibound levels close to the dissociation limit. Comparison with the results of fulldimensional calculations shows that a very good agreement is obtained when separating adiabatically the highfrequency H_{2}^{+} stretching motion from the remaining two lowfrequency modes and that further separation of the lowfrequency motions still provides useful approximations for the positions of the bound and lowest quasibound states.

13981404

Breakdown of the BornOppenheimer Approximation in InCl X^{1}^{+} Electronic State
by M. Molski and J. Konarski
Department of Theoretical Chemistry, Faculty of Chemistry, Adam Mickiewicz University, ul. Grunwaldzka 6, 60780 Poznan, Poland
(Received April 9th; revised June 6th, 1997; accepted June 15th, 1997)
Pure rotational transitions of ^{115}In^{35}Cl, ^{115}In^{37}Cl, ^{113}In^{35}Cl and^{ 113}In^{37}Cl were analyzed to yield massindependent radial parameters representing the internuclear potential energy and nonadiabatic rotational effects of the nuclei. From the determined t_{0}^{In,Cl} parameters, reflecting the purely the nonadiabatic rotational effects, and from experimental value of the dipole moment the rotational gfactor of four isotopic variants of InCl has been evaluated. The quality of the BornOppenheimer approximation and the extent of its breakdown in indium monochloride have been examined.

14051410

Full CI Calculations of Magnetic Properties of the H_{2} Molecule in the B^{1}_{u}^{+} State
by T. Helgaker^{1}, M. Jaszunski^{1,2} and K. Ruud^{1}
^{1}Department of Chemistry, University of Oslo, Box 1033, Blindern, N0315 Oslo, Norway
^{2}Institute of Organic Chemistry, Polish Academy of Sciences, 01 224 Warszawa, Kasprzaka 44, Poland
(Received February 2nd, 1997; revised March 11th, 1997; accepted March 11th, 1997)
The magnetizability, the gfactor, the NMR shielding constant and the spinrotation constant of the hydrogen molecule in the B^{1}_{u}^{+} state are studied in the full Configuration Interaction (full CI, FCI) approach. A GIAO orbital basis is used to ensure gaugeorigin indepedence of the computed properties. Their dependence on basis set and internuclear distance is analysed. Our results for the magnetizability confirm that this excited state of H_{2} is paramagnetic. The paramagnetism of this state is also reflected by the values of the other properties.

14111420

Applicability of the AlmostLinear CoupledCluster Method to Nondynamically Correlated States
by I. Grabowski, K. Jankowski and K. Kowalski
Institute of Physics, Nicholas Copernicus University, 87100 Torun, Poland
(Received April 14th, 1997; accepted July 8th, 1997)
To study the reliability of the almostlinear coupledcluster (ALCC) approach when applied to the description of states disclosing nondynamical correlation (or quasidegeneracy) effects, calculations have been performed for several systems widely used in quantumchemical model calculations for quasidegenerate states (H4 (MBS), H_{2}O (DZ), BeH_{2} (DZ), BH (DZP)). The ALCC approach represents a special application of the recently formulated splitamplitude strategy for approximating the equations of the CC methods (Chem.Phys. Lett., 256,141(1996)) which consist in splitting the individual cluster amplitudes into two components  one of fixed value and the other determined from a set of modified CC equations. The linearization of the latter set yields the equations of the ALCC method. In the present study the cluster operator is represented in terms of one and twobody operators (ALCCSD). The energies calculated are free from the singular behavior of their counterparts obtained for some strongly quasidegenerate states when using the standard linearized CC methods (LCCSD). Outside the very strong quasidegeneracy region the ALCCSD energies are very close to the CCSD ones, which demonstrates the usefulness of the low cost ALCC approach.

14211431

Radius of Convergence of the AmosMusher Intermolecular Perturbation Theory for the Hydrogen Molecule Ground State
by W.H. Adams
Wright and Rieman Chemistry Laboratories Rutgers University New Brunswick, NJ 08903 U.S.A. Email: adams@rutchem.rutgers.edu
(Received April 17th, 1997; revised June 18th, 1997; accepted June 19th, 1997)
We have approximately determined the radius of convergence of the AmosMusher perturbation theory applied to the hydrogen molecule at nuclear separations ranging from 3 to 12 bohr. We have done this by approximately locating for the lowest eigenvalue of the AmosMusher Hamiltonian those branch points which are closest to the center about which the perturbation expansion is developed. Using the same method and basis set we have also located the branch points of the Polarization Approximation applied to H_{2} and obtained results in good agreement with the accurate values found a few years ago by Cwiok, Jeziorski, Ko這s, et al. We find that the radius of convergence of the AmosMusher theory increases from 1.7 times as large as that of the Polarization Approximation at 3 bohr to twice as large at 8 to 12 bohr. This shows that the Amos Musher theory differs fundamentally from the Polarization Approximation.

14321446

Nonadditive Effects in Small Metal and Polyatomic Molecular Clusters
by O. Novaro
Instituto de Fisica, UNAM Apdo. Postal 20364, Mexico 01000, D.F., Mexico
(Received February 19th, 1997, revised January 5th, 1998; accepted January 10th, 1998)
A review of the role of multibody effects in the behavior of oligomer bound states, crystalline structures, adsorption properties of surfaces and dense states of matter is given. Special emphasis is placed on the work of W. Ko這s and coworkers in the Universities of Warsaw and Mexico. New results including electron correlation are given here, confirming the enormous role played by the multibody corrections in the stability of metallic clusters as Li_{n} and Be_{n}, where nonadditive effects are very large, to the point of making the convergence of the multibody expansion of the energy extremely doubtful. For polyatomic molecular clusters as (NH_{3})_{3} multibody effects albeit small, still are quite relevant to determine the cluster stability and geometry.

14471453

LongRange Potential for Excess Electron Surface States on Helium Clusters
by J. Jortner and M. Rosenblit
School of Chemistry, TelAviv University, Tel Aviv 69978, Israel
(Received October 2nd, 1997; accepted October 10th, 1997)
An excess electron bound to a (^{4}He)_{N} cluster (number of constituents N 3 x 10^{5} and cluster radius R R_{c} = 149 ) exhibits a divergence of the first moment of the charge distribution near the localization threshold (R = R_{c}), obeying the scaling law <r> (R  R_{c})^{1}. The asymptotic form of the electroncluster potential has to be modified to include CasimirSpruch retardation effects, which provide a small repulsive contribution O(/mc)r^{1}.

14541463

ManyBody Interactions, Symmetry Adapted Perturbation Theory and Chemical Bonding in Beryllium Clusters
by I. G. Kaplan
Instituto de Fisica UNAM, Apdo. Postal 20364, 01000 Mexico, D. F. MEXICO
(Received May 13th, 1997; revised June 15, 1997; accepted June 19, 1997)
The general formulae for the decomposition of manybody interaction energies in the double SAPT and the ab initio model potential for the Be_{3} cluster are presented. The analysis of exchange and dispersion contributions to the model potential allows to elucidate the nature of binding in beryllium clusters. Two kind of stabilizing forces are responsible for binding: the additive local van der Waals forces and the nonadditive delocalized exchange forces.

14641471

Dependence of the Solvation Free Energies on Atomic Charge Distribution
by P. Cieplak
Department of Chemistry, University of Warsaw, Pasteura 1, 02093 Warsaw, Poland
(Received April 14th, 1997; revised April 30th, 1997; accepted June 3th, 1997)
The dependency of hydration free energies on the atomic charge distribution has been studied using the free energy perturbation method and molecular dynamics simulations. Several hypothetical sets of atomic charges for methanol molecule have been constructed, yielding the same total molecular dipole moment of 2.14 D. Each of the sets has been perturbed into another one and the electrostatic contributions to the hydration free energy differences associated with such changes have been determined. The magnitude of calculated electrostatic contribution to the solvation free energies crucially depends on the position of the center of charge and orientation of the total dipole moment within molecular van der Waals envelope. It depends less on higher molecular multipole moments. The issue of convergence of the electrostatic contribution to the free energies obtained from molecular dynamics simulations will be also addressed.

14721478

Proton Affinities of Rare Gases
by M. Klobukowski
Department of Chemistry, University of Alberta, Edmonton, AB, T6G 2G2 Canada
(Received April 28th, 1997; accepted May 9th, 1997)
Density functional theory and large Gaussian basis sets were employed in the studies of the molecular ions HeH^{+}, NeH^{+}, and ArH^{+}. The calculated proton affinities of 42, 4951, and 9092 kcal/mol for the three rare gases He, Ne, and Ar agree reasonably well with the experimental values of 42, 49, and 88 kcal/mol. Spectroscopic constants for the ions were evaluated and compared with other accurate values.

14791496

Second Virial Coefficients for AtomMolecule Complexes from Ab Initio SAPT Potentials
by R. Moszynski^{1}, T. Korona^{1}, T.G.A. Heijmen^{2}, P.E.S.
Wormer^{2}, A. van der Avoird^{2}, and B. Schramm^{3}
^{1}Department of Chemistry, University of Warsaw, Pasteura 1, 02093 Warsaw, Poland ^{2}Institute of Theoretical Chemistry, NSR Center, University of Nijmegen, Toernooiveld 1, 6525 ED Nijmegen, The Netherlands ^{3}PhysikalischChemisches Institut, RuprechtKarlsUniversitaet Heidelberg, Im Neuenheimer Feld 500, D69120 Heidelberg 1, Germany
(Received May 6th, 1997; revised June 15th, 1997; accepted June 17th, 1997)
Ab initio potentials, calculated by symmetryadapted perturbation theory, are applied to compute second virial coefficients (including first order quantum corrections) for HeCO, NeCO, ArH_{2}, HeC_{2}H_{2}, and ArCH_{4} mixtures over a wide range of temperatures. The experimental methods are briefly outlined and the errors in the procedure to obtain mixed virial coefficients from the experimental data are discussed. Very good agreement with the majority of the available experimental data is observed for all systems except ArCH_{4} where calculated virial coefficients are slightly too high in comparison with measured data. Possible reasons for discrepancies between theory and experiment at very low temperatures are discussed.

14971504

Isoelectronic Dimers [(XH_{3})_{2}, (YH_{2})_{2}, (ZH)_{2}, and (Rg)_{2}] in the Groups of the Periodic System: Ab initio Quantum Chemical Calculations
by P. Hobza, J.V. Burda and R. Zahradnik
J. Heyrovsky Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, Dolejskova 3, 182 23 Prague 8, Czech Republic
(Received April 17th, 1997; accepted May 7th, 1997)
The (ZH)_{2}, (YH_{2})_{2}, (XH_{3})_{2} and (Rg)_{2} dimers [Z = FAt; Y = O, Po; X = N, Bi; Rg = rare gas] were studied ab initio using the CCSD(T) and MP2 procedures. Average relativistic effective potentials were used for all the halogens, while Stuttgart effective core potentials were used for the remaining nonhydrogen atoms. All the (HX)_{2} structures are Hbonded. All the stabilization energies mutually approach when passing down the group of the periodic system.

15051523

The Influence of the OH Stretch and O^{...}O Distance on the ManyBody Interactions in the Cyclic Water Trimer
by J. Rak^{a,b}, M.M. Szczesniak^{a}, G. Chalasinski^{c} and S.M. Cybulski^{d}
^{a}Department of Chemistry, Oakland University, Rochester, Michigan 48309, USA ^{b}Department of Chemistry, University of Gdansk, 80952 Gdansk, Poland ^{c}Department of Chemistry, University of Warsaw, Pasteura 1, 02093 Warsaw, Poland ^{d}Department of Chemistry and Biochemistry, Miami University, Oxford, Ohio 45056, USA
(Received June 4th, 1997; accepted June 17th, 1997)
The effects of the OH stretch and changes in the O^{...}O distance upon the one, two and threebody interaction energy components were studied in a cyclic water trimer. Calculations were carried out using supermolecular MoellerPlesset perturbation (SMPPT) theory in conjunction with perturbation theory of intermolecular forces, also known as intermolecular MoellerPlesset theory (IMPPT). The dependence of the onebody relaxation as well as electrostatic, induction, exchange and dispersion terms upon changes in the intramolecular OH bond length and upon the intermolecular O^{...}O distance were studied to identify the term which have the strongest influence on the structural parameters of the trimer. We found that the onebody terms must be evaluated at a high level of theory (MP4) with an elaborate basis set. The twobody perturbation energies appear to be reliably reproduced through the second order (MP2), and the threebody terms are reliable already at the self consistent field (SCF) level of theory. The twobody and threebody terms vary approximately linearly with the OH stretch of protondonors.

15241539

Hbridged Gas Phase Clusters of Methanol (Dimers to Hexamers): Ab Initio Calculations of their Structure and Vibrational Spectra
by J. Sauer and A. Bleiber
Max Planck Gesellschaft, Arbeitsgruppe "Quantenchemie" an der HumboldtUniversitaet Berlin, Jaegerstrasse 10/11, D10117 Berlin, Germany
(Received September 25th, 1997; revised November 27th, 1997; accepted November 30th, 1997)
HartreeFock calculations using a polarized basis set are performed on methanol clusters of two to six molecules. Electron correlation is included by second order MoellerPlesset perturbation theory for the trimer and the structure of the hexamer only. Starting from trimers, among several optimized structures the cyclic ones are found most stable. Harmonic vibrational frequencies and infrared intensities are calculated and comparison is made with observed spectra in the CO ad OH stretch region.

15401550

Periodic HartreeFock Studies on (HCl)_{} Chain
by S. Berski and Z. Latajka
Faculty of Chemistry, University of Wroclaw, 14 F. JoliotCurie, 50383 Wroclaw, Poland
(Received May 19, 1997; accepted June 10th, 1997)
The infinite chain of hydrogen chloride (HCl) _{
} is studied by the periodic HartreeFock (PHF) method using CRYSTAL 92 program. The basis sets of double zeta type containing sp diffuse as d and p polarisation functions are investigated. The relationship between structural and energetic parameters of (HCl)C and these basis sets is discussed. Furthermore, it is shown that [12s9p/6s4p] basis set of Veillard and Huzinaga, which was successfully used in studies on HCl dimer, yields the best geometry of the infinite chain. Based on this basis set, the height of the barrier for proton transfer is obtained with value of 39.6 kcal/mol. The comparison of the density of states plots (DOS) computed for (HF)_{
} and (HCl)_{} indicates that computational level influences rather slightly the DOS function in case of the hydrogen chloride chain.

15511564

Quantum Chemical Description of Catalytic Activation of the CH Bond
by E. Broclawik
Institute of Catalysis and Surface Chemistry, Polish Academy of Sciences, Cracow, Poland
(Received April 9th, 1997; revised June 30th, 1997; accepted July 1st, 1997)
Density functional theory is presented from the point of view of its applicability in modelling catalytic reactions. Reaction paths for CH bond scission in methane interacting with metal oxide/transition metal catalytic sites for gallium oxide in zeolitic lattice and palladium or rhodium based supported catalysts are analysed. On the basis of interdependence between the electronic structure and reactivity of the bond two mechanisms for the activation of the CH bond are described: via electrons withdrawal from bonding states (on oxide catalysts) and via electron insertion to antibonding orbitals (on metals).

15651583

Cluster Model Studies on Catalytic Properties of Vanadium Pentoxide
by M. Witko^{1}, R. Tokarz^{1} and K. Hermann^{2}
^{1}Institute of Catalysis and Surface Chemistry, Polish Academy of Sciences, ul. Niezapominajek, 30 239 Cracow, Poland ^{2}FritzHaberInstitut der MaxPlanckGesellschaft, Faradayweg 46, 14195 Berlin, Germany
(Received April 8th, 1997, accepted May 10th, 1997)
ZINDO cluster model calculations are performed to study the electronic structure and chemical reactivity of the V_{2}O_{5}(010) surface. Interatomic binding in vanadium pentoxide is determined to be of a mixed ionic and covalent character. The calculations reveal the difference in the catalytic properties between structurally inequivalent surface oxygen centers and show the increased local reactivity of bridging oxygens with respect to the electrophilic adparticles. Convergence of the electronic properties with respect to the cluster size is achieved for cluster as large as V_{10}O_{31}H_{12}. The effect of the second substrate layer on the surface electronic properties is found to be negligible. Further, rather similar electronic parameters of the V_{10}O_{31}H_{12} cluster in its idealized, bulk and optimized geometry are obtained. The H/H^{+} species adsorb at the V_{2}O_{5}(010) surface, always at oxygen sites, forming very stable surface hydroxyl groups. The strongest binding occurs with the oxygen O(c) bridging two bare vanadium atoms. These O(c) oxygens become quite mobile in presence of the H/H^{+} adparticle. Allowing the surface oxygens to relax during adsorption of H/H^{+} leads to different adsorption scenarios depending on the surface oxygen site. At the vanadyl oxygen site a very stable and rigid hydroxyl group O(a)H is formed above the vanadium center. At the doubly coordinated oxygen site O(b) the adsorbate penetrates between two vanadyl groups to form a local O(b)H group, while at the O(c) site an O(c)H group is created (slightly above the surface O(c) position), where OH binding is strongest. Weak initial interaction, between triply coordinated oxygens O(d,e) and the incoming H/H^{+} species, leads to stabilization of the adsorbate near the closest vanadyl site resulting in a tilted O(a)H group instead of O(d)H or O(e)H.

15841592

Low Frequency Vibrations in Electroabsorption Spectroscopy
by M. Slawik and P. Petelenz
Department of Theoretical Chemistry, Jagiellonian University, 30060 Krakow, Ingardena 3, Poland
(Received April 9th, 1997; accepted May 30th, 1997)
A simple twostate, twomode model is proposed to probe the role of lowfrequency vibrations in electroabsorption (EA) spectroscopy. The results demonstrate that explicit inclusion of such vibrations in the model is necessary to reproduce the EA signals of Frenkel states, but essentially irrelevant for CT states.

15931603

Carbyne Trefoil Knots  Are They Stable Enough to Exist?
by J.Cz. Dobrowolski^{1,2} and A.P. Mazurek^{2}
^{1}Industrial Chemistry Research Institute, 8 Rydygiera Street, 01793 Warsaw, Poland ^{2}Drug Institute,3034 Chelmska Street, 00725 Warsaw, Poland
(Received March 27th, 1997; revised June 6th, 1997; accepted June 15th, 1997)
The structure and stability of nonlinear carbon clusters of carbyne type was estimated based on ab initio quantum mechanical calculations. Both the cyclic (cycarbynes) and knotted (knocarbynes) trefoil structures of closed polycarbon chains, containing up to 84 carbon atoms, were considered. Comparison of the calculated spectral properties (IR, ^{13}C NMR) with those of C_{60} fullerene provide the basis for experimental identification of C_{n} cyclic and knotted structures as may have implication for interstellar dust spectroscopy.

16041614

Electron Affinity of CH_{3} and BH_{3} and the Structure of their Anions
by G.L. Gutsev and R.J. Bartlett
Quantum Theory Project, P.O. Box 118435, University of Florida, Gainesville, FL 326118435
(Received April 29th, 1997; revised June 3th, 1997, accepted June 30th, 1997)
The electronic and geometrical structure of the methyl radical, CH_{3}, the borane molecule, BH_{3}, as well as their anions CH_{3}^{} and BH_{3}^{} are calculated with the HartreeFock Density Functional Theory (HFDFT), fourthorder manybody perturbation theory (MBPT(4)), and the coupledcluster (CCSD(T)) methods with the use of the large atomic natural orbital basis of WidmarkMalmqvistRoos. Potential energy surfaces of BH_{3} and BH_{3}^{} symmetric stretchings and outofplane C_{3nu} hydrogen motions of CH_{3} and CH_{3}^{} are obtained at the MBPT(4)/6311++G(3df, 3pd) level. At the CCSD(T) level of theory, the difference in the total electronic energies of CH_{3} and CH_{3}^{} is 0.01 eV and that of BH_{3} and BH_{3}^{} is 0.05 eV. Corrected for the zeropoint energies (ZPE) of nuclear motions, the adiabatic electron affinities (EA_{ad}) of CH_{3} and BH_{3} are 0.05 and 0.01 eV, respectively, being in nice agreement with the experimental values of 0.08±0.03 and 0.038±0.015, respectively. Thus, the EA_{ad}s of CH_{3} and BH_{3} are defined mainly by the differences in the ZPEs of the corresponding neutralanion pairs.

16151623

AbInitio Theoretical Study of DipoleBound Anions of Molecular Complexes. Water Molecule Inhibits or Enhances Electron Affinity of NMethylaminoadenine
by J. Smets, D.M.A. Smith, Y. Elkadi and L. Adamowicz
Department of Chemistry, University of Arizona, Tucson, Arizona 85721, U.S.A.
(Received April 28th, 1997, accepted May 30th, 1997)
Ab initio calculations have been performed to determine the electron affinity of the water complex of Nmethylaminoadenine (NMA). This complex has been chosen for this study because it can form three different isomeric structures with a water molecule hydrogenbonded at three different NMA sites, and only two of these structures have sufficient dipole moments to form dipolebound states with an excess electron. In the third structure, which according to the calculations should be the most stable form, the dipole moments of NMA and the water molecule oppose each other resulting in an almost null dipole moment of the complex, so the complex has no ability to form a stable dipolebound anion. This determination offers an interesting possibility for an experimental investigation of whether the NMA.H_{2}O complex forms an stable anion. If such an anion is detected, it will mean that an electron attachment can increase the gas phase concentration of the complex, which is thermodynamically less stable in its neutral form.

16241629

GIAOCHF and Experimental Study of the Substituent Effects on ^{13}C Magnetic Shielding in Benzene Derivatives
by K. Jackowski^{1}, A. Les^{1}, A. Dambska^{1} and L. Adamowicz^{2}
^{1}Department of Chemistry, The University of Warsaw ul. Pasteura 1, 02093 Warszawa, Poland
^{2}Department of Theoretical Chemistry, The University of Lund, Chemical Center, P.O.B. 124, S22100 Lund, Sweden
(Received March 15th, 1997; revised May 5th, 1997; accepted May 6, 1997)
Ab initio calculations of the ^{13}C shielding constants were performed with the use of the GIAOCHF method for the benzene molecule and its 14 monosubstituted derivatives. The theoretical substituent effects were compared with our experimental data which were measured in cyclohexane solutions and extrapolated to infinite dilution. Satisfactory correlations between theoretical and experimental results were observed for all the aromatic carbons except those in the meta position.

16301634

The Existence of Two Isomeric Forms of Monofluorinated Hydrogen Sulfide: A New Theoretical Rationale for Experimentally Observed Infrared Spectrum
by P. Babinec^{1,2} and J. Leszczynski^{1}
^{1}Department of Chemistry, Jackson State University, P.O. Box 17910, Jackson, MS 39217, USA
^{2}Department of Chemical Physics, Comenius University, Bratislava, Slovakia
(Received March 13th, 1997; accepted April 4th, 1997)
In order to explain the discrepancies between the matrix isolated IR spectra of the products of the H_{2}S + F_{2} reaction and recent computational studies of this system, we have analyzed the potential energy surface of monofluorinated hydrogen sulfide using high level postHartreeFock methods. The molecular geometries and harmonic vibrational frequencies were calculated at the DFT, MP2, and CCSD levels in conjunction with the 6311++G(d,p) basis set. Besides the already proposed HSF isomer which corresponds to the global energetic minimum, a local minimum corresponding to the HFS isomer as well as a transition state structure between these minima have been also characterized. The calculated vibrational frequencies fit well with the experimental IR spectra and also complement previous ab initio calculations on the HSF isomer.

16351656

Dipole Moment and Polarizability Functions of Ammonia: A LinearResponse CoupledCluster Study
by P. Piecuch^{1}, V. Spirko^{2} and J. Paldus^{3}
^{1}Department of Chemistry, University of Toronto, 80 St. George Street, Toronto, Ontario, Canada M5S 3H6 ^{2}J. Heyrovsk<203> Institute of Physical Chemistry, Academy of Sciences of the Czech Republic, Dolejskova 3, 18223 Prague 8, Czech Republic ^{3}Quantum Theory Group, Department of Applied Mathematics, and Department of Chemistry and (GWC)^{2}  Waterloo Campus, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1
(Received August 20th, 1997; revised September 25th, 1997; accepted October 1th, 1997)
The recently developed orthogonally spinadapted linearresponse coupledcluster theory with singly and doubly excited clusters (LRCCSD) has been employed to calculate the dipole moment and parallel polarizability functions of ammonia over a wide range of values of the inversional and symmetric stretch coordinates. Using the eigenstates of the experimentally derived nonrigid invertor Hamiltonian of ammonia and the LRCCSD property functions, the transition dipole moments between several lowlying inversional states and the rovibrationally averaged parallel polarizabilities for several values of the inversional quantum number have been evaluated. The calculated transition dipole moments and rovibrationally averaged polarizabilities have been found to be in excellent agreement with the available experimental and theoretical data, indicating the suitability of the ab initio LRCCSD theory for such calculations. The LRCCSD transition dipole moments are far more accurate than all previously obtained ab initio values.

16571666

Prediction of Polymers with Several Optimal Physical Properties
by J.J. Ladik
Chair for Theoretical Chemistry and Laboratory of the National Foundation of Cancer Research of the FriedrichAlexander University ErlangenNuremberg, Egerlandstr. 3, D91058 Erlangen, Germany
(Received June 4th, 1997; accepted July 4th, 1997)
First a brief review is given of the ab initio HartreeFock and correlation corrected band structure calculation methods of periodic 1D and 2D polymers. In the 1D case, the extension of the theory to disordered chains leading to the calculation of variable range hopping conductivity of some native proteins is also outlined. In the cases of the ground state properties of (SN)_{x}, for the vibrational and excitonic spectra of organic  and biopolymers good agreement could be obtained with experiment. The same has been achieved for the fundamental gap of different organic polymers, for the bulk modulus of polyethylene and for the hopping conductivity along their main chains of insulin and lysozyme. These examples demonstrate that if one applies sophisticated enough theoretical metohods, any kind of physical property of any kind of periodic or nonperiodic quasi1D polymer can be computed in good agreement with experiment. This opens up the possibility to predict polymers with optimal 3 5 nonrelated properties from a family of polymers with a huge number of members. The prediction of such "tailormade" polymers is of course of large practical importance. It is discussed that such a theoretical approach to find polymers with 45 optimal properties is much less expensive (and it will be still less expensive in the future) than the classical procedure to synthetize and measure the properties of a larger number of polymers.

16671679

Ab initio Protein Folding  Is Conformational Space Searching Enough?
by K.A. Olszewski
Molecular Simulations Inc., 9685 Scranton Rd., San Diego, CA 92121, USA
(Received May 21th, 1997; accepted May 30th, 1997)
Ab initio protein folding is a common name for protein structure prediction approaches, that explore the conformational space of a protein using a model of a protein and a simple, carefully designed, potential energy function. Attempts to predict the native state of the protein or to reproduce the folding pathway from a set of simple rules are essential for an understanding of the physicochemical rules that govern the folding process. Noteworthy, a number of new techniques for protein structure/function prediction that widen the search space and the meaning of the potential energy function have emerged. In this paper the variety of approaches to protein structure or function prediction are discussed and classified with respect to their distribution in the general protein sequencestructure space.

16801690

Transient Cavities in Liquids and the Nature of the Hydrophobic Effect
by A. Pohorille^{1,2}
^{1}Exobiology Branch, NASA  Ames Research Center MS 2394 Moffett Field, California 940351000
^{2}Department of Pharmaceutical Chemistry, University of California, San Francisco, California 94143
(Received June 15th, 1997; revised July 30th, 1997; accepted August 15th, 1997)
The size distributions of transient cavities in water and organic liquids, obtained from computer simulations, have provided a new means to analyze the nature of the hydrophobic effect and to evaluate the adequacy of different analytical models of this effect. The poor solubility of nonpolar solutes in water is attributed to a low probability of finding in water cavities of atomic and molecular size. It has been shown that water applies more force per unit area of cavity surface than do hydrocarbon liquids. Models that successfully capture the main characteristics of the hydrophobic effect must at least include information about the density and the radial distribution of oxygen atoms in liquid water. One such model, quantitatively accurate for molecular solutes of arbitrary shape, is presented.

16911736

On the Hierarchy of electron Models
by L.Z. Stolarczyk
Department of Chemistry, University of Warsaw, Pasteura 1, 02093 Warsaw, Poland
(Received November 17th, 1997; accepted November 26th, 1997)
A systematic derivation of electron models is given, based on a spinrestricted formulation of the molecularorbital (MO) model for closed and highspin openshell states. A chain of stepbystep approximations leads to the following topdown hierarchy of MO models: ab initio ( separation) PariserParrPople (PPP) Hubbard ,ßselfconsistent Hueckel Molecular Orbital (HMO) ßselfconsistent HMO "basic" HMO. The PPP semiempirical parameters ('s, ß's, and 's) are related to exact MO formulas, and the limitations of the PPP MO model are discussed. In general, the parameters of a given
electron model are related to those of the preceding one, and limitations of the introduced approximations are indicated. It is shown, for instance, that for the closedshell alternant hydrocarbons the ßselfconsistent HMO model, applied for describing the properties of molecular groundstate (equilibrium geometry, thermochemical stability), is equivalent to the more advanced Hubbard model.

17371746

Variational Calculation of the Global Hardness and the Fukui Function via an Approximation of the Hardness Kernel
by F. De Proft^{1}, P. Geerlings^{1}, S. Liu^{2} and R. G. Parr^{2}
^{1}Enheid Algemene Chemie, Vrije Universiteit Brussel, Falculteit Wetenschappen, Pleinlaan 2, 1050 Brussels, Belgium
^{2}Department of Chemistry, University of North Carolina, Chapel Hill, North Carolina 275993290, USA
(Received May 20th, 1997; accepted July 4th, 1997)
An implementation of the variational principle for the chemical hardness [P.K. Chattaraj, A. Cedillo and R.G. Parr, J. Chem. Phys., 103, 7645(1995)] is described, which uses a simple approximation to the hardness kernel. The molecular Fukui function is taken to be a linear combination of atomic Fukui functions, which are expanded in a set of Gaussian type orbitals. This methodology is shown to yield results which are largely in agreement with sensitivity analysis methods of Mortier, Nalewajski, and coworkers.

17471762

Density Functional Method  Principles and Applications
by J. Andzelm
Molecular Simulations, Inc., 9685 Scranton Rd, San Diego, CA 92121
(Received May 30th, 1997, accepted July 4th, 1997)
An overview of the main concepts of Density Functional Theory from a historical perspective is given. Most popular techniques of solving DFT equations currently used in chemistry are listed. Applications of the DFT method to study chemical reactions using various levels of DFT theory are discussed. Local DFT is successful in predicting molecular structures with covalent bonds. However, in general it should not be used to study energetics of reactions and systems with weak interactions. The next level of theory, nonlocal DFT, is a preferable choice for the study of thermochemistry. Investigation of reaction barriers may require a higher level of theory, so called hybrid methods that use a portion of the exact HartreeFock exchange.

17631778

On the Chemical Potential/Electronegativity Equalization in Density Functional Theory
by R.F. Nalewajski
K. Guminski Department of Theoretical Chemistry, Jagiellonian University, R. Ingardena 3, 30060 Cracow, Poland
(Received March 7th, 1997; accepted March 20th, 1997)
General variational principles of the KohnSham (KS) density functional theory are interpreted as the corresponding chemical potential/electronegativity equalization equations. The unconstrained (groundstate) and constrained (excited) electron configurations are examined for both the system global description and for the case of its partitioning into mutually closed subsystems, e.g., reactants. The chemical potential discontinuity for the integer numbers of electrons at zero temperature is stressed, and the KS orbital description of the charge transfer (CT) between reactants is discussed. Using the appropriate ensemble formulation of the KS theory the in situ chemical potential/electronegativity difference, the driving "force" behind the interreactant CT, is linked to the relevant KS frontier eigenvalues of polarized reactants.

17791791

Exploring Bonding Patterns of Molecular Systems Using Quantum Mechanical Bond Multiplicities
by J. Mrozek^{1}, R.F. Nalewajski^{1} and A. Michalak^{2}
^{1}K. Guminski Department of Theoretical Chemistry, Jagiellonian University, R. Ingardena 3, 30060 Cracow, Poland ^{2}Department of Computational Methods in Chemistry, Jagiellonian University, R. Ingardena 3, 30060 Cracow, Poland
(Received March 27th, 1997; accepted June 30th, 1997)
One of the fundamental chemical concepts, the structural formula of a chemical species, which visualizes bonding patterns in a given molecular system, can also be given a more flexible and quantified form in terms of alternative quantummechanical measures of bondorders. The recently developed difference approach, based upon the onedeterminantal wavefunction [HartreeFock (HF) and KohnSham (KS) theories], compares the chargeandbondorder matrix in a molecule (P) and that in the separated atoms limit (P^{0}, SAL) defined in the orthogonal basis set of atomic functions. In this approach the atomic and diatomic contributions to the molecularly averaged difference between these matrices, <P>/2 = 1/2tr[P(P^{0}  P)], are used to define bond multiplicities. The main purpose of this work is to examine the overall performance of such KS and HF bonding indices, when applied to the set of chemically well defined reference systems, and to use them to investigate bondorders in more challenging molecules, which have recently been the subject of independent studies. Manifestations of changing bond ionicity by these effective bondorders are examined for a series of diatomics. Some methodological aspects associated with alternative SAL choices are discussed and tested.

17921797

Theoretical Density Functional Studies of Tautomeric and Conformational Forms of N^{4}hydroxycytosine and Its 5fluoro and 5methyl Derivatives
by G. Bakalarski^{1,2} and B. Lesyng^{2,3}
^{1}Department of Biophysics, Agriculture University, Rakowiecka 26, 02528 Warsaw, Poland ^{2}Interdisciplinary Centre for Mathematical and Computational Modelling, Pawinskiego 5A, 02106 Warsaw, Poland ^{3}Department of Biophysics, Warsaw University, Zwirki i Wigury 93, 02089 Warsaw, Poland
(Received May 20th, 1997, accepted July 4th, 1997)
Density functional (DFT) calculations at the BLYP/DNP and B3LYP/631G(d,p) levels were performed for amino and imino tautomers of N^{4}hydroxycytosine (HC), N^{4}hydroxy5fluorocytosine (5FHC) and N^{4}hydroxy5methylcytosine (5MeHC). Full optimization of the molecular skeleton and zeropoint vibrational corrections computed at the B3LYP level were included in the energy analysis. The results are compared with ab initio MBPT(2)/631G(d,p)//HF/631G(d,p) calculations [1] performed for HC and 5FHC. The DFT methods, and in particular the B3LYP approximation, predict properly the higher stability of the imino tautomers, and with lower computational costs reproduce very well the energy differences obtained using the conventional ab initio methods.

17981825

Electron Localization in Liquid Methanol. Quantum PathIntegral Simulation
by W.M. Bartczak ^{1,2} and M. Sopek^{1,}
^{1}Institute of Applied Radiation Chemistry, Technical University of Lodz, Lodz, Wroblewskiego 15, Poland ^{2}Department of Theoretical Chemistry, University of Lodz, Lodz, Pomorska 149/153, Poland
(Received April 18th, 1997; revised June 26th, 1997, accepted July 4th, 1997)
An excess electron in liquid methanol at room temperature was studied using the method of PathIntegral Molecular Dynamics simulation. A compact charge distribution of an excess electron, suggesting a localized electron state, was found. The charge distribution is centred in a cavity built of methanol according to the traditional picture of the solvated electron. Various radial distribution functions were calculated reflecting the correlations between the cavity centre or the electron charge density and the sites of the methanol molecule. The correlations are stronger than in the case of the hydrated electron. Interpretation of the radial distribution functions as well as the bondangle distribution functions leads to a picture of 4 methanol molecules forming the solvation shell of the solvated electron. The molecules are oriented towards the centre of the electron density by the OH bonds. The coordination number of the solvated electron agrees with the conclusions from electron magnetic resonance experiments.

18261848

Multiple Electron Exchanges in Calculations of Pairwise Nonadditive Contribution to Trimer Interaction Energy
by V.F. Lotrich^{a}, K. Szalewicz^{a} and B. Jeziorski^{b}
^{a}Department of Physics and Astronomy, University of Delaware, Newark, DE 19716 ^{b}Department of Chemistry, University of Warsaw, Pasteura 1, 02093 Warsaw, Poland
(Received August 26th, 1997; revised October 7th, 1997; accepted October 7th, 1997)
The first order of the symmetryadapted perturbation theory has been used to compute the pairwise nonadditive contribution to the exchange repulsion energy in the He_{3}, Ne_{3}, Ar_{2}HF, and (H_{2}O)_{3} trimers. The contributions from single and double electron exchanges, as well as from the cyclic permutations involving three electrons at a time, were calculated separately and compared with the accurate result involving all possible electron permutations. The intramonomer electron correlation was completely neglected and the HartreeFock determinants were used to represent the monomer wave functions. The importance of the resulting threebody contributions, which can be quadratic (single exchanges), cubic (threeelectron cycles), and quartic (double exchanges) in a typical intermolecular overlap integral S, is studied for several geometrical configurations of the investigated systems. Whereas the S^{2} and S^{3} terms are important for all systems, the role of the S^{4} terms turns out to be very small for Ar_{2}HF, large for the raregas timers at linear configurations, and appreciable for the water trimer. The higher than double exchanges (S^{5} and higher terms) are negligible for all systems except for the water trimer at very small intermonomer separations where the expansion in powers of S ceases to be useful. The nonadditivity of the socalled zerothorder exchange energy, i.e., the difference between, the HeitlerLondon and the firstorder energies, has also been investigated. The zerothorder exchange energy contains only the S^{4} and higher terms and becomes significant when the S^{4} contribution to the firstorder energy is important. The S^{2} and S^{3} contributions to the firstorder exchange nonadditivity in the Ar_{2}HF trimer agree well with the calculations of Cybulski and collaborators [J. Chem. Phys., 101, 10708 (1994), ibid., 106, 3301 (1997)] using their pseudodimer model.

18491857

Multiple EllipticalGaussianDensity Annealing as a Tool for Finding the Most Stable Structures. Application to LennardJones Atomic Clusters
by J. Pillardy and L. Piela
Quantum Chemistry Laboratory, Department of Chemistry, University of Warsaw, 02093 Warsaw, Pasteura 1, Poland
(Received April 10th, 1997; accepted April 15th, 1997)
Smoothing of the potential energy hypersurface is a promising way to reduce complexity of the original hypersurface, thus facilitating the search for the most stable configuration of a molecular system. Despite of the effort made in the last decade to find an efficient smoothing technique, a reliable and economical method is still sought. One of the powerful approaches is the Gaussian Density Method (GDA) of Straub and coworkers. In the method one solves the reduced Bloch equation that describes the evolution of the spatial part of the canonical density distribution, when the temperature changes. In the GDA method this distribution is assumed, for each atom, as a single threedimensional isotropic Gaussian with the position and width changing according to known equations of motion, when the temperature changes. In the present paper we allow for a threedimensional elliptical Gaussian distribution for each atom. Additionally, when in the course of lowering the temperature the anisotropy of the ellipse becomes large enough, the single Gaussian distribution for an atom may branch into two elliptical Gaussian distributions. Evolution in temperature of the new distributions for the system is calculated by solving for each of them the independent reduced Bloch equation. Finally, when the temperature reaches 0 K, one has a number of Gaussian distributions, each corresponding to a structure and (usually low) energy of the system. The method has been applied to the clusters of N argon atoms (N = 5,..., 33), the system serving usually as benchmark. Allowing for the anisotropy of the Gaussian distributions results in a remarkable increase of numerical stability.
